Warning! Warning! And a reading recommendation
Hi Pete,
I don't think the description you quoted is so awful for a unit on universal gravitation, so I'd advise you not to get all worried :-/
pibomb said:
The author of your book seems to misunderstand that the spacetime of general relativity is flat.
Oh my. I am surprised that no-one (apparently) has yet objected to this statement, since it is, if you will pardon the pun, flat out wrong. At least, under the reasonable assumption that by "flat" pibomb means "vanishing Riemann curvature tensor".
(After discarding several possible guesses about what pibomb could have been thinking, it finally occurred to me that perhaps he/she was thinking of the -Ricci- curvature tensor. However, a four dimensional Lorentzian manifold with vanishing Ricci tensor but nonvanishing Riemann tensor is generally called "Ricci flat", or in the context of gtr, "a vacuum solution". However, even this modified claim would only describe correctly -vacuum- solutions, not spacetime models including some dust, an electromagnetic field, or whatever.)
Pedantic quibbles: the Minkowski spacetime is indeed flat and it is trivially a vacuum solution to the Einstein field equation (EFE), and even curved spacetimes might have nonzero curvature in some region (for example, a model of a thin uniform density spherical shell), so it is probably best to say that "spacetime models in gtr generally have nonzero Riemann curvature at most events".
parlyne said:
The "warping space and time" comment is exactly true. The "lumpy" thing, however just doesn't make any sense.
Parlyne, I think you are being rather harsh! I agree with stingray: while the quotation from the textbook was taken out of context, if for example you are discussing something like a galaxy full of main sequence stars, on a suitable length scale, "lumpy" might not give such a bad first approximation to the general idea of how gtr treats gravitation.
Jeff Reid said:
My understanding is that gravity is an attractive force and that it also affects times (as gravitational field strength increases, time slows down).
Well, saying that "time slows down" makes no sense at all, when you think about it, and of course gtr says no such thing. Rather, it represents the gravitational field as the curvature of spacetime itself. A fundamental feature of curved manifolds is the fact that a pair of initially parallel geodesics (analogue of straight lines in euclidean space) may diverge or converge as you run along the pair. To understand what gtr really says about "gravitational time dilation", let's consider a simple thought experiment:
Let us imagine two observers in two rocket ships who use their rocket engines to hover motionless at two radii r_1, \, r_2 where r_2 > r_1 > 0, over a massive object. Suppose that the closer observer is sending out time signals every second (by his clock) to the farther observer. You can picture the "world lines" of these signals as "null geodesics". As it happens, the vacuum field equation of gtr dictates in this situation that radially outgoing null geodesics must diverge, so that the farther observer finds that according to HIS clock, the signals are arriving at intervals longer than one second.
Another manifestation of essentially the same effect is the so-called "gravitational red-shifting" of light signals: if our close observer fires a laser with frequency \nu at the farther observer, its frequency as measured by the farther observer will be smaller than expected.
Notice that time certainly has not slowed down anywhere in this picture!
If you are curious to learn more, you might enjoy the excellent popular books by Robert Geroch and by Robert Wald (both leading experts on gtr work in the same department at the University of Chicago, as it happens) listed at ttp://www.math.ucr.edu/home/baez/RelWWW/reading.html#pop
Chris Hillman
